Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \frac{\cos ^{3} x-\sin ^{3} x}{\cos x-\sin x}=1+\sin x \cos x $$

Step-by-Step Solution

Answer

The identity is verified by factoring and simplifying, then canceling common terms.

Step 1: Use the difference of cubes formula

Recognize that the numerator \( \cos^3 x - \sin^3 x \) can be factored as a difference of cubes. The difference of cubes formula is \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \). Here, set \( a = \cos x \) and \( b = \sin x \). This gives: \( (\cos x - \sin x)(\cos^2 x + \cos x \sin x + \sin^2 x) \).

Step 2: Simplify using the Pythagorean identity

Recall the Pythagorean identity \( \cos^2 x + \sin^2 x = 1 \). Substitute into the expression \( \cos^2 x + \cos x \sin x + \sin^2 x = 1 + \cos x \sin x \).

Step 3: Cancel common terms

Since the numerator \( (\cos x - \sin x)(1 + \cos x \sin x) \) and the denominator \( (\cos x - \sin x) \) share a common factor, we can cancel \( (\cos x - \sin x) \) from the numerator and denominator, resulting in \( 1 + \cos x \sin x \).

Step 4: Verify identity

After cancellation, we are left with \( 1 + \cos x \sin x \), which is equal to the right side of the given identity. This confirms that the initial expression \( \frac{\cos ^{3} x-\sin ^{3} x}{\cos x-\sin x} \) simplifies to \( 1 + \cos x \sin x \), therefore verifying the identity.

Key Concepts

Difference of Cubes Formula

The difference of cubes formula is a valuable tool in algebra, allowing us to factor expressions of the form \( a^3 - b^3 \). It states that:

• \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \)

This formula is useful when dealing with cubic expressions and seeking to simplify or manipulate them.
To understand this better, let's consider it geometrically. Imagine two cubes with edges of length \( a \) and \( b \). The difference of their volumes \( a^3 - b^3 \) can be "broken down" into components that align with

• the edge difference \( (a-b) \)
• and the remaining quadratic polynomial \( (a^2 + ab + b^2) \).

For trigonometric functions, this formula helps us factor expressions such as \( \cos^3 x - \sin^3 x \). We treat \( \cos x \) as \( a \) and \( \sin x \) as \( b \), allowing us to simplify the expression. Incorporating this factorization is key in our step-by-step solution approach to verify trigonometric identities.

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry. It arises from the Pythagorean Theorem applied to the unit circle.
At the heart of trigonometry, this identity states that:

• \( \cos^2 x + \sin^2 x = 1 \).

This equation holds true for any angle \( x \). The reason it is so important is because it helps in simplifying trigonometric expressions.
Consider the expression derived in our problem \( \cos^2 x + \cos x \sin x + \sin^2 x \). Using the Pythagorean identity, we can substitute \( 1 \) for \( \cos^2 x + \sin^2 x \), transforming the expression into \( 1 + \cos x \sin x \).
Such substitutions are vital in verifying and proving trigonometric identities, offering a way to synthesize complex calculations into simpler, more manageable forms.

Trigonometric Simplification

Trigonometric simplification is an essential skill when dealing with identities, equations, or expressions involving trigonometric functions. The aim is to reduce complex forms to simpler ones, often by cancelling common factors or applying identities.
In our exercise, we started with the expression:

• \( \frac{\cos^3 x - \sin^3 x}{\cos x - \sin x} \)

By factoring the numerator using the difference of cubes, we obtained:

• \( (\cos x - \sin x)(1 + \cos x \sin x) \)

Simplifying this involves recognizing and cancelling the common factor \( \cos x - \sin x \) from both the numerator and the denominator.
Once the common term is cancelled, the expression reduces to a much simpler form \( 1 + \cos x \sin x \), which was the target of our verification task.
This process showcases the utility of simplification in trigonometry, making complex expressions clearer and more concise, thus aiding in solving and verifying problems effectively.